isuckatmath wrote:Let u and v be (fixed) vectors in the vector space V. Show that the set W of all linear combinations ab+bv of u and v is a subspace of V.

Show that generic elements of the set fulfill the definition of a vector space. (There should be a list of properties for vector spaces. You need to show that these linear combinations obey these rules.)

isuckatmath wrote:Prove: If the (finite) set S of vectors contains the zero vector, then S is linearly dependent.

What is the definition of a linearly independent set? If you add the zero vector to such a set, what property no longer holds?

isuckatmath wrote:Determine whether or not the given vectors in Rn form a basis for Rn.v1=(3,-7,5,2), v2=(1,-1,3,4), v3=(7,11,3,13)

Since the vectors are in R4 and you have only three vectors (so clearly they cannot form a basis), try to find a vector in R4 which is not in the span of the set you've been given.

isuckatmath wrote:Let {v1,v2,...,vk} be the basis for the proper subspace W of the vector space V, and suppose that the vector v of V is not in W. Show that the vectors v1,v2,...,vk,v are linearly independent.

What is the definition of a basis? So what can you say about the vectors in the basis?

If the vector v is not in W, can you form v by a linear combination of the basis vectors? So what can you say about the set with this new vector thrown in?